A steel wire of area of cross-section A and length 2L is clamped firmly between two points seperated by a distance .2L.. A body is hung from the middle point of the wire such that the middle point sags by a distance x. Calculate the mass of the body and the angle made by string with the horizontal

Explanation

Since .`theta`. is small
`sin theta = tan theta=(x)/(L)`
`y=(F)/(A).(L)/(e )`
`F=(YA e)/(L)=(YA)/(L)[(L^(2)+x^(2))^(1//2)-L]`
`F=(YA)/(L)[L(1+(x^(2))/(2L^(2)))-L]=(YA)/(L)[L+(x^(2))/(2L)-L]`
`F=(YA x^(2))/(2L^(2)) , 2T sin theta=Mg`
`2(T)theta=Mg" "(because " for small angles "sin theta=theta)`
`2Ftheta =Mg " " 2.(YA x^(2))/(2L^(2))theta=Mg`
`(2YA x^(2))/(2L^(2)). (x)/(L)=Mg, " "M=(YAX^(3))/(L^(3)g)`
`(x)/(L)=((Mg)/(YA))^(1//3), "Tan" theta=(x)/(L)`
`"Tan"theta=((Mg)/(YA))^(1//3) rArr theta ="Tan"^(-1)((Mg)/(YA))^(1//3)`